non-linear codes for asymmetric channels, applied to optical channels

Non-Linear Codes for Asymmetric Channels, applied to Optical Channels

Miguel Griot

Miguel Griot Oral Qualifying Exam - UCLA - Electrical Engineering

2

Outline Motivation : Optical Channel, Uncoordinated Multiple Access. Models and Capacity Calculation

Basic Model: the OR Channel Treating other users as noise

Capacity loss vs. complexity reduction. The Z channel

The need for non-linear codes Optimal ones density

Non-linear Trellis Coded Modulation (NL-TCM) Definition of distance in the Z-Channel Characteristics of Trellis Codes Design Technique Results for 6-user OR-MAC & 100-user OR-MAC Concatenation with High-Rate Block Codes

Results for 6-user OR-MAC Conclusions Future Work


3

Motivation: Optical Channels, Multiple Access Optical Channels:

provide very high data rates, up to tens to hundreds of gigabits per second.

Typically deliver a very low Bit Error Rate Wavelength Division (WDMA) or Time Division (TDMA)

are the most common forms of Multiple Access today. However, they require considerable coordination.

Objective Uncoordinated access to the channel. Apply error correcting codes, in order to achieve the

required BER. Maximizing the rate at feasible complexity for optical

speeds.

9( 10 )BER


4

Basic Model: The OR Multiple Access Channel (OR-MAC) OR Channel model

Basic model that can describe the multiple-user optical channel with non-coherent combining

N users transmitting at the same time

If all users transmit a 0, then a 0 is received

If even one of them transmits a 1, a 1 is received

0+X=X, 1+X=1

User 1

User 2

User N

Receiver


5

OR Channel: Theoretical characteristics Achievable rate (Capacity):

The theoretical limits for the MAC, were given by Liao and Ahslwede.

In the case of the OR-MAC, the Theoretical Capacity is the triangle of all rate-pairs less than the maximum possible sum-rate, which is 1.

This sum-rate can be theoretically achieved by: Joint Decoding. Sequential decoding (requires

coordination). Time-Sharing or Wave-length sharing

(requires coordination).


6

Treating other users as noise: the Z-Channel Joint Decoding and Successive Decoding are fully

efficient in that one useful bit of information is transmitted per time-wavelength slot.

However, non of these are computationally feasible for optical speeds today.

A practical alternative is to treat all but a desired user as noise.

This alternative, while dramatically reducing the decoding complexity, looses up to 30% of full capacity, as we will see next.

When treating other users as noise in an OR-MAC, each user “sees” what is called the Z-Channel.

My research has been focused on the Z-Channel, resulting from the OR-MAC when treating other users as noise.


7

The Z-Channel N users, all transmitting with the same ones

density p: P(X=1)=p, P(X=0)=1-p.Focus on a desired user

If it transmits a 1, a 1 will be received. If it transmits a 0, a 0 will be received only if all

other N-1 users transmit a 0

iX

0

1

0

111 (1 )Np

1(1 )Np

YiXp

1 p


8

Maximum achievable sum-rate, when treating other users as noise. Information Theory tells us the optimal ones density

to transmit for each user. When the number of users tends to infinity, the

optimal ones density tends to , which is also the optimal density for joint decoding.

In that case equal probabilities of 1 and 0 is perceived at the receiver.

Note that for a large number of users, the optimal ones density becomes very small.

Surprisingly, the maximum achievable sum-rate is always lower-bounded by ln(2)=0.6931 and tends to ln(2) when the number of users tends to infinity.

Np /1)2/1(1


9

Comparison of capacities

2 3 4 5 6 7 8 9 10 11 120

0.2

0.4

0.6

0.8

1

Number of users

Cap

acity

Sum rate comparison

Other users as noiseJoint decodingp1 = 0.5

ln(2)

Optimal ones densities:

Users Joint Others noise

2 0.293 0.286

6 0.109 0.108

12 0.056 0.056


10

The need for non-linear codes Linear codes provide equal density of ones and zeros in their output

(p=0.5). Most of the codes thoroughly studied in the literature are linear

codes. We observed in previous slide that, for linear codes, the achievable

rate tends to zero as the number of users increase. As the number of users increase, the optimal ones density tends to

zero. Non-linear codes with relatively low density of ones are

required, to a achieve a good rate. Only recently, there has been work on LDPC codes with arbitrary

density of ones. Theoretical bounds are found to prove that these codes are capacity achieving under ML decoding. There is no design technique described for these codes.

Non-linear Trellis Coded Modulation This work introduces a novel design technique for non-linear trellis

codes with an arbitrary density of ones. To my knowledge, it is the first work that addresses this task.


11

Interleaver Division Multiple Access One successful approach to uncoordinated multiple

access is IDMA. Every user has the same channel code, but each user’s

code bits are interleaved by a randomly drawn interleaver, with very high probability of being unique.

The receiver is assumed to know the interleaver of the desired user.

With IDMA in the OR-MAC, a receiver should see the signal from a desired user, corrupted by a memory-less Z-Channel.

Performance obtained for a 6-user OR-MAC using IDMA, and for the corresponding Z-Channel were the same in my simulations.


12

Trellis Codes Characteristics Memory given by a state. In the trellis representation, for each state, and each

possible input, an output value and the next state is given.

Generally next state and output given by generator polynomials. Initial state: the all-zero state. Zero Termination. They are NOT capacity achieving

We are achieving around 30% of full capacity (around 43% of the achievable rate when treating other users as noise)

Low complexity compared to capacity achieving codes (Turbo-Codes, LDPC) ML decoding: Viterbi Decoding

1 2 0, , ,v vX X X

(0) (0)1 0, ,vX X 0:010

1:100

State at time t:State at time (t+1):

(1) (1)1 0, ,vX X


13

Metric for the Z-Channel, for Maximum Likelihood decoding Given a received word, the decoded codeword will be the one that

maximizes , or given equally likely codewords . For the Z-Channel, if one codeword has a 1 in a position where

the received word r has a 0, then

Among the possible transmitted codewords (where there are no 1-to-0 transitions):

where is the number of 0-to-1 transitions, and is the number of 0-to-0 transitions

Note that for the possible transmitted codewords is actually the number of zeros in the received word, which is the same for all possible codewords.

Now, , so the most likely codeword is the one that presents the less number of 0-to-1 transitions.

|ip c r

| 0kp r c kc

| ip r c

( )01 00| (1 )iN N

ip r c

1

( )01iN 00N

00N


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Definition of distance for the Z-Channel The distance between two codewords measures the

likelihood that one transmitted codeword will be wrongly decoded as the other codeword.

In the Z-Channel, a transmitted 1 will always induce a received 1.

Define the directional Hamming distance as the number of ones that have to be added to a codeword so that all ones of codeword are present in the received word.

Example:

Now:

1 2( , )Dd c c

1c 2c

1 21 1 2

2 2 1 2 1

111011001001 ( , ) 3110011 111011 ( , ) 1

D

D

rc d c cc r d c c

• A Maximum-Likely (ML) decoder will always decode the codeword with larger Hamming weight.

1 201 013, 1N N


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Definition of distance for the Z-Channel (2) For two codewords with different Hamming weight, if the

received word contains all ones from both codewords, the one with larger Hamming weight will be more likely than the codeword with smaller Hamming weight .

Only if is transmitted, an error will be produced in the decoder.

Then, the directional distance of interest is which is the larger of both directional Hamming distances.

For two codewords with equal Hamming weight, errors can be made in both directions, and both directional Hamming distances are equal, and equal to the maximum of both.

In any case, the proper pairwise deign metric is:

And the overall objective is to maximize:

1 2( , )Dd c c

, , max , , ,i j j i D i j D j id c c d c c d c c d c c

2c

1c1c

min max , , ,i j D i j D j id c c d c c


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Greedy definition of distance In a trellis code, the design is made branch-wise: for

each state, and each input, we assign the next state, and the output.

Due to its non-linearity, last definition cannot be applied branch-wise.

It is impossible to tell from one branch, which codeword will have more Hamming weight.

Hence, we have to consider both branch-wise directional Hamming distances.

The safest branch-wise metric would be:

This is the definition used in our design of NL-TCM. , , min , , ,i j j i D i j D j id c c d c c d c c d c c


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Non-linear Trellis Coded Modulation Desired density of ones p is given Rate of the form: 1/n (1 input bit, n output bits). states (represented by v bits) 2S branches Feed-forward encoder with 1 input:

Design: Assign output values to the 2S branches of the trellis Objective: Maximize the minimum distance (“greedy definition”) Those outputs have to maintain the desired density of ones p.

2vS

1 2 0, , ,v vX X X 2 0, , ,0vX X

2 0, , ,1vX X

0

1

State at time t:State at time (t+1):


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Assigning Hamming Weights First step: assign Hamming weights to the output of

each branch. Using any of the definitions of distance given before,

codewords with as equal Hamming weight between each other lead to better performance.

In the case of codewords with different Hamming weights, the worst-case performance will be driven by those codewords with smaller Hamming weight.

Criteria: assign as similar Hamming weights to the branches as possible, maintaining the density of ones as close to the desired density of ones as close to the desired p as possible.


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Assigning Hamming Weights Consider the following sub-graph:

There are S/2 of these sub-graphs. Branches produced by an input bit equal to 0 for both

states (or 1) go to the same state. Define

In this subgroup of four branches, assign a Hamming weight of w+1 to i branches, and a Hamming weight of w to (4-i) branches.

2 00, , ,vX X

2 01, , ,vX X

2 0, , ,0vX X

2 0, , ,1vX X

0

01

1

( )(4( ))

w floor p ni round p n w


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Assigning Hamming Weights, Examples: 6-user OR-MAC, desired density of ones is .

n=20 : w=2, i=2 2 branches with Hw=2, 2 with Hw=3 (p=1/8).

n = 18 : w=2, i=1 3 branches with Hw=2, 1 with Hw=3 (p=1/8).

n = 17 : w=2, i=round(0.5) 1 branch with Hw=3 and 3 with Hw=2 (p=0.132) all with Hw=2 (p=2/17=0.118).

100-user OR-MAC, n = 400 : w=2, i=3 (p = 0.006875) n = 360 : w=2, i=2 (p = 0.006944)

1/8p

0069.0p


21

Choosing all branches to have at least distance of 1 between each other It would be desirable if possible, that all branches had at least

distance of 1 between each other. In the case where all branches have the same Hamming weight w

then we can have up to different branches with a distance of at least 1.

If , it is possible.

In the case of branches with different Hamming weights, the computation is a little more complicated.

Two different codewords split at some point in their trellis paths, and their paths will not merge again until at least v+1 trellis sections after the split.

In case it is possible to have different output values for all branches, then the minimum distance of the code is lower-bounded by

nw

2n

Sw

min 1d v


22

Choosing all branches to have at least distance of 1 between each other In case we need to repeat output values, we can allow the following

branches to have same output value maintaining the bound Again: consider the sub-graph

Branches in red (blue) can have same output value without affecting the minimum distance.

Consider two different paths, one traversing branch A, and the other traversing branch B at some trellis section. They traverse at least v branches with different output values before

that trellis section. They traverse at least v branches with different output values after that

trellis section. Their distance is at least 2v.

min 1d v

2 00, , ,vX X

2 01, , ,vX X

2 0, , ,0vX X

2 0, , ,1vX X

0

01

1A

B


23

Ungerboeck’s rule We have already assigned Hamming weights to the branches, and have

enumerated all the possible output values in order to have different output values for all branches (allowing some branches to be equal according to previous slide)

Up to this point we have We can further increase the minimum distance by applying Ungerboeck’s

rule: maximize the distance between all splits and merges.

Remember that all output values had at least a Hamming distance of w. For every two different codewords, their paths split and merge at least once,

and there are at least v-1 branches between the split and the merge. Hence:

min 1d v

min 2 1d w v

2 00, , ,vX X

2 01, , ,vX X

2 0, , ,0vX X

2 0, , ,1vX X

split

split

2 00, , ,vX X

2 01, , ,vX X

2 0, , ,0vX X

2 0, , ,1vX X

merge

merge


24

Extending Ungerboeck’s rule One can extend Ungerboeck’s rule

into the trellis.0

1

Maximize split


25


into the trellis.0

1

Maximize

0

1

0

1


26

Note that by maximizing the distance between the 8 branches, coming from a split 2 trellis section before, we are maximizing all groups of 4 branches coming from a split in the previous trellis section, and all splits.


into the trellis.0

1Maximize

0

1

0

1


27

Extending Ungerboeck’s rule The same idea applies for the merges,

moving backwards in the trellis. If we move h trellis sections forward from a

split (including the split), and g sections backwards from a merge (including the merge), the new bound becomes:

Now, we have to compute the maximum possible values of h and g.

min ( ) 1 ( )d w h g v h g


28

Extending Ungerboeck’s rule First, let’s compute the number of branches

that need to have maximum distance between each otherto cover h sections from a split, and g sections backwards from a merge.

From a splitting point of view: From the merging point of view: Each branch, belongs to one group of

and one group of Thus, each branch has to have maximum

distance with other branches.

2h2g

2h 2g

2 2 2h g


29

Extending Ungerboeck’s rule Second, we can compute how many branches of

maximum distance between each other we can have. Let’s denote this number T.

For branches with equal Hamming weight w, In the general case

The constraints are: Each branch has to belong to a group of and a

group of If we choose all constraints are satisfied

( / )T floor n w

4 / 4 / 4 ( 1)nT floor

i w i w

2 ,2h gT T

2 2 2h g T 2g

2h


30

Designing for a very low desired ones density For a low enough desired ones density, all the branches can be

chosen to have maximum distance. The design becomes straight-forward. Consider a NL-TCM code with S states, desired density p. Denote M the sum of all the ones from the outputs of all 2S

branches. Then: But if then It is possible to choose all 2S branches so that there is at most 1

branch that has a 1 in a given position. Straight-forward design:

Assign Hamming weights to branches For each branch, add ones in positions that aren’t used in

previous branches Example: 100-user OR-MAC,

)(2 npSM

Sp

21

nM

0078.0)2/(10069.0 Sp


31

Performance Results For all implementations, states were used. 6-user OR-MAC

n=20 : Sum-rate = 0.30 2 branches with Hw=2, 2 with Hw=3 (p=1/8). h=3, g=2 :

n = 18 : Sum-rate = 1/3 3 branches with Hw=2, 1 with Hw=3 (p=1/8). h=2,g=2 :

n = 17 : Sum-rate = 0.353 all with Hw=2 (p=2/17=0.118). h=2,g=2 :

100-user OR-MAC, n = 400 : w=2, i=3 (p = 0.006875) n = 360 : w=2, i=2 (p = 0.006944) for both cases

62 64

min 2 5 (6 1) 5 12d

min 11d

min 11d

min 14d


32

Performance results FPGA implementation:

In order to prove that NL-TCM codes are feasible today for optical speeds, a hardware simulation engine was built on the Xilinx Virtex2-Pro 2V20 FPGA.

Results for the rate-1/20 NL-TCM code are shown next.

Transfer Bound: Wen-Yen Weng collaborated to this work, with the

computation a Transfer Function Bound for NL-TCM codes.

It proved to be a very accurate bound, thus providing a fast estimation of the performance of the NL-TCM codes designed in this work.


33

Performance Results : 6-user OR-MAC

4 5 6 7 8

10-6

10-5

10-4

10-3

users

BER

NL-TCM 1/17NL-TCM 1/18NL-TCM 1/20


34


0.2 0.3 0.4 0.5 0.6 0.7

10-8

10-6

10-4

10-2

100

BER

NL-TCM 1/17NL-TCM 1/18NL-TCM 1/20NL-TCM 1/20 FPGABound 1/17Bound 1/18Bound 1/20


35

Results: observations An error floor can observed for the

FPGA rate-1/20 NL-TCM. This is mainly due to the fact that, while

theoretically a 1-to-0 transition means an infinite distance, for implementation constraints those transitions are given a value of 20.

Trace-back depth of 35. The BER for all cases are not as low

as required.


36


Rate Sum-rate p BER

1/360 0.2778 0.006944 0.49837

1/400 0.25 0.006875 0.49489

64.54 10

79.45 10


37

Dramatically lowering the BER : Concatenation with Outer Block Code Optical systems deliver a very low BER, in our work a

was required. Using only a NL-TCM, the rate would have to be very low. A better solution is found using the fact that when the

Viterbi decoding fails, with relatively high probability only a small number of bits are in error.

Thus, a high-rate block code that can correct a few errors can be attached as an outer code, dramatically lowering the BER.

910BER

Block-Code Encoder NL-TCM Encoder

Z-ChannelBlock-Code Decoder NL-TCM Decoder


38

Reed-Solomon + NL-TCM : Results A concatenation of the rate-1/20 NL-TCM code with (255

bytes,247 bytes) Reed-Solomon code has been tested for the 6-user OR-MAC scenario.

This RS-code corrects up to 8 erred bits. The resulting rate for each user is (247/255).(1/20) The results were obtained using a C program to apply the

RS-code to the FPGA NL-TCM output.

Although we don’t have results for the 100-user case, it may be inferred that a similar BER would be achieved.

Rate Sum-rate p BER

0.0484 0.29 0.125 0.4652

102.48 10


39

Conclusions I developed a novel design technique for non-linear

trellis codes, that provide a wide range of ones density. These codes have been designed for the Z-Channel, that

arises in the optical multiple access channel. A relatively low ones density is essential for the OR-MAC

channel, and asymmetric channels in general. An arbitrary number of users is supported, maintaining

relatively the same efficiency (around 30%) Although these codes are not capacity achieving,a good

part of the capacity is achieved, with a suitable BER fr optical needs, and a complexity feasible for optical speeds with today’s technology. An FPGA implementation has been built to prove this fact.


40

Future work (1): Capacity achieving codes Capacity achieving codes. Although they may not be feasible for optical speeds,

with today’s technology, Turbo codes and LDPC codes will be feasible in the near future

Part of my immediate future’s work will be the design Turbo-Like codes, with an arbitrary ones density.

Most common Turbo-like codes are Parallel concatenation of convolutional codes Serially concatenated convolutional codes.

The convolutional codes will be replaced by properly designed NL-TCMs.


41

Non-linear Turbo Like codes Serial concatenation CC + NL-TCM:

Parallel concatenated NL-TCMs:

CC Interleaver NL-TCM

NL-TCM

Interleaver NL-TCM


42

Non-linear Turbo-like codes The NL-TCM will not be a feed-forward

encoder. The design criteria changes. However, the fundamental ideas hold. The fact that the RS+NL-TCM concatenation

(hard-decision transmitted from one decoder to the other) has such a good BER, makes the serial concatenation of CC+NL-TCM with soft-decoding look promising.


43

Future Work(2): More general Channel Also, to be more general, I will study the Multiple

access channel where the 1+1=0 case, has a positive (although very small) probability.

Treating other users as noise, one user “sees” an Binary Asymmetric Channel.

This will be change the metric in the Viterbi decoder, the definition of distance used, but shouldn’t change the design criteria

0

1

0

1

YiXp

1 p1

1

non-linear codes for asymmetric channels, applied to optical channels

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